3.188 \(\int \frac{\cosh ^6(x)}{a+b \sinh (x)} \, dx\)

Optimal. Leaf size=145 \[ -\frac{a x \left (20 a^2 b^2+8 a^4+15 b^4\right )}{8 b^6}-\frac{2 \left (a^2+b^2\right )^{5/2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b^6}+\frac{\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac{\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac{\cosh ^5(x)}{5 b} \]

[Out]

-(a*(8*a^4 + 20*a^2*b^2 + 15*b^4)*x)/(8*b^6) - (2*(a^2 + b^2)^(5/2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]]
)/b^6 + Cosh[x]^5/(5*b) + (Cosh[x]^3*(4*(a^2 + b^2) - 3*a*b*Sinh[x]))/(12*b^3) + (Cosh[x]*(8*(a^2 + b^2)^2 - a
*b*(4*a^2 + 7*b^2)*Sinh[x]))/(8*b^5)

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Rubi [A]  time = 0.414462, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2695, 2865, 2735, 2660, 618, 206} \[ -\frac{a x \left (20 a^2 b^2+8 a^4+15 b^4\right )}{8 b^6}-\frac{2 \left (a^2+b^2\right )^{5/2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b^6}+\frac{\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac{\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac{\cosh ^5(x)}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^6/(a + b*Sinh[x]),x]

[Out]

-(a*(8*a^4 + 20*a^2*b^2 + 15*b^4)*x)/(8*b^6) - (2*(a^2 + b^2)^(5/2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]]
)/b^6 + Cosh[x]^5/(5*b) + (Cosh[x]^3*(4*(a^2 + b^2) - 3*a*b*Sinh[x]))/(12*b^3) + (Cosh[x]*(8*(a^2 + b^2)^2 - a
*b*(4*a^2 + 7*b^2)*Sinh[x]))/(8*b^5)

Rule 2695

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + p)), x] + Dist[(g^2*(p - 1))/(b*(m + p)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] &&
NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2865

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c*(m + p + 1) -
a*d*p + b*d*(m + p)*Sin[e + f*x]))/(b^2*f*(m + p)*(m + p + 1)), x] + Dist[(g^2*(p - 1))/(b^2*(m + p)*(m + p +
1)), Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1
) - d*(a^2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2,
0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*m]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cosh ^6(x)}{a+b \sinh (x)} \, dx &=\frac{\cosh ^5(x)}{5 b}+\frac{i \int \frac{\cosh ^4(x) (-i b+i a \sinh (x))}{a+b \sinh (x)} \, dx}{b}\\ &=\frac{\cosh ^5(x)}{5 b}+\frac{\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}-\frac{i \int \frac{\cosh ^2(x) \left (i b \left (a^2+4 b^2\right )-i a \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{4 b^3}\\ &=\frac{\cosh ^5(x)}{5 b}+\frac{\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac{\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac{i \int \frac{-i b \left (4 a^4+9 a^2 b^2+8 b^4\right )+i a \left (8 a^4+20 a^2 b^2+15 b^4\right ) \sinh (x)}{a+b \sinh (x)} \, dx}{8 b^5}\\ &=-\frac{a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac{\cosh ^5(x)}{5 b}+\frac{\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac{\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac{\left (a^2+b^2\right )^3 \int \frac{1}{a+b \sinh (x)} \, dx}{b^6}\\ &=-\frac{a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac{\cosh ^5(x)}{5 b}+\frac{\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac{\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac{\left (2 \left (a^2+b^2\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b^6}\\ &=-\frac{a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac{\cosh ^5(x)}{5 b}+\frac{\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac{\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}-\frac{\left (4 \left (a^2+b^2\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac{x}{2}\right )\right )}{b^6}\\ &=-\frac{a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}-\frac{2 \left (a^2+b^2\right )^{5/2} \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{b^6}+\frac{\cosh ^5(x)}{5 b}+\frac{\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac{\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}\\ \end{align*}

Mathematica [C]  time = 5.7785, size = 857, normalized size = 5.91 \[ \frac{\cosh (x) \left (240 (a+i b)^2 \sqrt{b^2} \tanh ^{-1}\left (\frac{\sqrt{a-i b} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}}}{\sqrt{a+i b} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}}}\right ) \sqrt{i \sinh (x)+1} (a-i b)^3+\sqrt{a+i b} \left (24 \sqrt{a-i b} b^4 \sqrt{b^2} \sqrt{i \sinh (x)+1} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}} \sinh ^4(x)-30 a \sqrt{a-i b} b^3 \sqrt{b^2} \sqrt{i \sinh (x)+1} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}} \sinh ^3(x)+8 \sqrt{a-i b} \left (b^2\right )^{3/2} \left (5 a^2+11 b^2\right ) \sqrt{i \sinh (x)+1} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}} \sinh ^2(x)-15 a \sqrt{a-i b} b \sqrt{b^2} \left (4 a^2+9 b^2\right ) \sqrt{i \sinh (x)+1} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}} \sinh (x)+2 \sqrt{b^2} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}} \left (4 \sqrt{a-i b} \left (15 a^4+35 b^2 a^2+23 b^4\right ) \sqrt{i \sinh (x)+1} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}}-15 (-1)^{3/4} \sqrt{b} \left (8 a^4-4 i b a^3+16 b^2 a^2-7 i b^3 a+8 b^4\right ) \sin ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{a-i b} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}}}{\sqrt{b}}\right )\right )-240 i \sqrt{a-i b} b \left (a^2+b^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{-i b} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}}}{\sqrt{i b} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}}}\right ) \sqrt{i \sinh (x)+1}\right )\right )}{120 \sqrt{a-i b} \sqrt{a+i b} b^5 \sqrt{b^2} \sqrt{i \sinh (x)+1} \sqrt{-\frac{b (\sinh (x)-i)}{a+i b}} \sqrt{-\frac{b (\sinh (x)+i)}{a-i b}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^6/(a + b*Sinh[x]),x]

[Out]

(Cosh[x]*(240*(a - I*b)^3*(a + I*b)^2*Sqrt[b^2]*ArcTanh[(Sqrt[a - I*b]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))])/(
Sqrt[a + I*b]*Sqrt[-((b*(-I + Sinh[x]))/(a + I*b))])]*Sqrt[1 + I*Sinh[x]] + Sqrt[a + I*b]*((-240*I)*Sqrt[a - I
*b]*b*(a^2 + b^2)^2*ArcTan[(Sqrt[(-I)*b]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))])/(Sqrt[I*b]*Sqrt[-((b*(-I + Sinh
[x]))/(a + I*b))])]*Sqrt[1 + I*Sinh[x]] - 15*a*Sqrt[a - I*b]*b*Sqrt[b^2]*(4*a^2 + 9*b^2)*Sqrt[1 + I*Sinh[x]]*S
inh[x]*Sqrt[-((b*(-I + Sinh[x]))/(a + I*b))]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))] + 8*Sqrt[a - I*b]*(b^2)^(3/2
)*(5*a^2 + 11*b^2)*Sqrt[1 + I*Sinh[x]]*Sinh[x]^2*Sqrt[-((b*(-I + Sinh[x]))/(a + I*b))]*Sqrt[-((b*(I + Sinh[x])
)/(a - I*b))] - 30*a*Sqrt[a - I*b]*b^3*Sqrt[b^2]*Sqrt[1 + I*Sinh[x]]*Sinh[x]^3*Sqrt[-((b*(-I + Sinh[x]))/(a +
I*b))]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))] + 24*Sqrt[a - I*b]*b^4*Sqrt[b^2]*Sqrt[1 + I*Sinh[x]]*Sinh[x]^4*Sqr
t[-((b*(-I + Sinh[x]))/(a + I*b))]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))] + 2*Sqrt[b^2]*Sqrt[-((b*(-I + Sinh[x])
)/(a + I*b))]*(-15*(-1)^(3/4)*Sqrt[b]*(8*a^4 - (4*I)*a^3*b + 16*a^2*b^2 - (7*I)*a*b^3 + 8*b^4)*ArcSin[((1/2 +
I/2)*Sqrt[a - I*b]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))])/Sqrt[b]] + 4*Sqrt[a - I*b]*(15*a^4 + 35*a^2*b^2 + 23*
b^4)*Sqrt[1 + I*Sinh[x]]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))]))))/(120*Sqrt[a - I*b]*Sqrt[a + I*b]*b^5*Sqrt[b^
2]*Sqrt[1 + I*Sinh[x]]*Sqrt[-((b*(-I + Sinh[x]))/(a + I*b))]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))])

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Maple [B]  time = 0.037, size = 674, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^6/(a+b*sinh(x)),x)

[Out]

2/b^6/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2))*a^6+6*a^4/b^4/(a^2+b^2)^(1/2)*arctanh
(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2))+6*a^2/b^2/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b
^2)^(1/2))-1/5/b/(tanh(1/2*x)-1)^5-1/2/b/(tanh(1/2*x)-1)^4+1/5/b/(tanh(1/2*x)+1)^5-1/2/b/(tanh(1/2*x)+1)^4+2/(
a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2))+13/12/b/(tanh(1/2*x)+1)^3-9/8/b/(tanh(1/2*x)
+1)^2+15/8/b/(tanh(1/2*x)+1)-13/12/b/(tanh(1/2*x)-1)^3-9/8/b/(tanh(1/2*x)-1)^2-15/8/b/(tanh(1/2*x)-1)-1/2/b^3/
(tanh(1/2*x)+1)^2*a^2+1/b^5/(tanh(1/2*x)+1)*a^4-1/2/b^4/(tanh(1/2*x)+1)*a^3+1/4/b^2/(tanh(1/2*x)+1)^4*a+1/3/b^
3/(tanh(1/2*x)+1)^3*a^2-1/2/b^2/(tanh(1/2*x)+1)^3*a-1/2/b^4/(tanh(1/2*x)-1)*a^3-1/4/b^2/(tanh(1/2*x)-1)^4*a-1/
3/b^3/(tanh(1/2*x)-1)^3*a^2-1/2/b^2/(tanh(1/2*x)-1)^3*a-1/2/b^4/(tanh(1/2*x)-1)^2*a^3-1/2/b^3/(tanh(1/2*x)-1)^
2*a^2-1/b^5/(tanh(1/2*x)-1)*a^4+1/2/b^4/(tanh(1/2*x)+1)^2*a^3-5/2/b^3/(tanh(1/2*x)-1)*a^2-9/8/b^2/(tanh(1/2*x)
-1)*a+5/2*a^3/b^4*ln(tanh(1/2*x)-1)+15/8*a/b^2*ln(tanh(1/2*x)-1)+11/8/b^2/(tanh(1/2*x)+1)^2*a+5/2/b^3/(tanh(1/
2*x)+1)*a^2-9/8/b^2/(tanh(1/2*x)+1)*a-5/2*a^3/b^4*ln(tanh(1/2*x)+1)-15/8*a/b^2*ln(tanh(1/2*x)+1)-11/8/b^2/(tan
h(1/2*x)-1)^2*a-a^5/b^6*ln(tanh(1/2*x)+1)+a^5/b^6*ln(tanh(1/2*x)-1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^6/(a+b*sinh(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.92581, size = 3822, normalized size = 26.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^6/(a+b*sinh(x)),x, algorithm="fricas")

[Out]

1/960*(6*b^5*cosh(x)^10 + 6*b^5*sinh(x)^10 - 15*a*b^4*cosh(x)^9 + 15*(4*b^5*cosh(x) - a*b^4)*sinh(x)^9 + 10*(4
*a^2*b^3 + 7*b^5)*cosh(x)^8 + 5*(54*b^5*cosh(x)^2 - 27*a*b^4*cosh(x) + 8*a^2*b^3 + 14*b^5)*sinh(x)^8 - 120*(a^
3*b^2 + 2*a*b^4)*cosh(x)^7 + 20*(36*b^5*cosh(x)^3 - 27*a*b^4*cosh(x)^2 - 6*a^3*b^2 - 12*a*b^4 + 4*(4*a^2*b^3 +
7*b^5)*cosh(x))*sinh(x)^7 - 120*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^5 + 60*(8*a^4*b + 18*a^2*b^3 + 11*b
^5)*cosh(x)^6 + 20*(63*b^5*cosh(x)^4 - 63*a*b^4*cosh(x)^3 + 24*a^4*b + 54*a^2*b^3 + 33*b^5 + 14*(4*a^2*b^3 + 7
*b^5)*cosh(x)^2 - 42*(a^3*b^2 + 2*a*b^4)*cosh(x))*sinh(x)^6 + 15*a*b^4*cosh(x) + 2*(756*b^5*cosh(x)^5 - 945*a*
b^4*cosh(x)^4 + 280*(4*a^2*b^3 + 7*b^5)*cosh(x)^3 - 1260*(a^3*b^2 + 2*a*b^4)*cosh(x)^2 - 60*(8*a^5 + 20*a^3*b^
2 + 15*a*b^4)*x + 180*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x))*sinh(x)^5 + 6*b^5 + 60*(8*a^4*b + 18*a^2*b^3 +
11*b^5)*cosh(x)^4 + 10*(126*b^5*cosh(x)^6 - 189*a*b^4*cosh(x)^5 + 48*a^4*b + 108*a^2*b^3 + 66*b^5 + 70*(4*a^2*
b^3 + 7*b^5)*cosh(x)^4 - 420*(a^3*b^2 + 2*a*b^4)*cosh(x)^3 - 60*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x*cosh(x) + 90
*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x)^2)*sinh(x)^4 + 120*(a^3*b^2 + 2*a*b^4)*cosh(x)^3 + 20*(36*b^5*cosh(x)
^7 - 63*a*b^4*cosh(x)^6 + 28*(4*a^2*b^3 + 7*b^5)*cosh(x)^5 + 6*a^3*b^2 + 12*a*b^4 - 210*(a^3*b^2 + 2*a*b^4)*co
sh(x)^4 - 60*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^2 + 60*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x)^3 + 12*(
8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x))*sinh(x)^3 + 10*(4*a^2*b^3 + 7*b^5)*cosh(x)^2 + 10*(27*b^5*cosh(x)^8 -
54*a*b^4*cosh(x)^7 + 28*(4*a^2*b^3 + 7*b^5)*cosh(x)^6 - 252*(a^3*b^2 + 2*a*b^4)*cosh(x)^5 + 4*a^2*b^3 + 7*b^5
- 120*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^3 + 90*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x)^4 + 36*(8*a^4*b
+ 18*a^2*b^3 + 11*b^5)*cosh(x)^2 + 36*(a^3*b^2 + 2*a*b^4)*cosh(x))*sinh(x)^2 + 960*((a^4 + 2*a^2*b^2 + b^4)*c
osh(x)^5 + 5*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^4*sinh(x) + 10*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^3*sinh(x)^2 + 10*(
a^4 + 2*a^2*b^2 + b^4)*cosh(x)^2*sinh(x)^3 + 5*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^4 + (a^4 + 2*a^2*b^2 +
b^4)*sinh(x)^5)*sqrt(a^2 + b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh
(x) + a*b)*sinh(x) - 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) +
2*(b*cosh(x) + a)*sinh(x) - b)) + 5*(12*b^5*cosh(x)^9 - 27*a*b^4*cosh(x)^8 + 16*(4*a^2*b^3 + 7*b^5)*cosh(x)^7
- 168*(a^3*b^2 + 2*a*b^4)*cosh(x)^6 - 120*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^4 + 72*(8*a^4*b + 18*a^2*
b^3 + 11*b^5)*cosh(x)^5 + 3*a*b^4 + 48*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x)^3 + 72*(a^3*b^2 + 2*a*b^4)*cosh
(x)^2 + 4*(4*a^2*b^3 + 7*b^5)*cosh(x))*sinh(x))/(b^6*cosh(x)^5 + 5*b^6*cosh(x)^4*sinh(x) + 10*b^6*cosh(x)^3*si
nh(x)^2 + 10*b^6*cosh(x)^2*sinh(x)^3 + 5*b^6*cosh(x)*sinh(x)^4 + b^6*sinh(x)^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**6/(a+b*sinh(x)),x)

[Out]

Timed out

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Giac [B]  time = 1.22981, size = 389, normalized size = 2.68 \begin{align*} \frac{6 \, b^{4} e^{\left (5 \, x\right )} - 15 \, a b^{3} e^{\left (4 \, x\right )} + 40 \, a^{2} b^{2} e^{\left (3 \, x\right )} + 70 \, b^{4} e^{\left (3 \, x\right )} - 120 \, a^{3} b e^{\left (2 \, x\right )} - 240 \, a b^{3} e^{\left (2 \, x\right )} + 480 \, a^{4} e^{x} + 1080 \, a^{2} b^{2} e^{x} + 660 \, b^{4} e^{x}}{960 \, b^{5}} - \frac{{\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \, b^{6}} + \frac{{\left (15 \, a b^{4} e^{x} + 6 \, b^{5} + 60 \,{\left (8 \, a^{4} b + 18 \, a^{2} b^{3} + 11 \, b^{5}\right )} e^{\left (4 \, x\right )} + 120 \,{\left (a^{3} b^{2} + 2 \, a b^{4}\right )} e^{\left (3 \, x\right )} + 10 \,{\left (4 \, a^{2} b^{3} + 7 \, b^{5}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-5 \, x\right )}}{960 \, b^{6}} + \frac{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac{{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right |}}\right )}{\sqrt{a^{2} + b^{2}} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^6/(a+b*sinh(x)),x, algorithm="giac")

[Out]

1/960*(6*b^4*e^(5*x) - 15*a*b^3*e^(4*x) + 40*a^2*b^2*e^(3*x) + 70*b^4*e^(3*x) - 120*a^3*b*e^(2*x) - 240*a*b^3*
e^(2*x) + 480*a^4*e^x + 1080*a^2*b^2*e^x + 660*b^4*e^x)/b^5 - 1/8*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x/b^6 + 1/96
0*(15*a*b^4*e^x + 6*b^5 + 60*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*e^(4*x) + 120*(a^3*b^2 + 2*a*b^4)*e^(3*x) + 10*(4
*a^2*b^3 + 7*b^5)*e^(2*x))*e^(-5*x)/b^6 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*log(abs(2*b*e^x + 2*a - 2*sqrt(a
^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^6)